Digital Signal Processing/Windowing

Most digital signals are infinite, or sufficiently large that the dataset cannot be manipulated as a whole. Sufficiently large signals are also difficult to analyze statistically, because statistical calculations require all points to be available for analysis. In order to avoid these problems, engineers typically analyze small subsets of the total data, through a process called windowing.

Windowing Introduction
Windowing is the process of taking a small subset of a larger dataset, for processing and analysis. A naive approach, the rectangular window, involves simply truncating the dataset before and after the window, while not modifying the contents of the window at all. However, as we will see, this is a poor method of windowing and causes power leakage.

Applying Windows
Application of a window to a dataset will alter the spectral properties of that dataset. In a rectangular window, for instance, all the data points outside the window are truncated and therefore assumed to be zero. The cut-off points at the ends of the sample will introduce high-frequency components.

Consider the system H(z), with input X(z) and output Y(z). We model this as:


 * $$Y(z) = X(z) H(z)$$

If we have a window with transfer function W(z), we can mathematically apply the window to our signal, X(z) as such:


 * $$\hat{X}(z) = X(z) W(z)$$

Then, we can pass our windowed signal into our system, H(z) as usual:


 * $$\hat{Y}(z) = \hat{X}(z) H(z)$$

Leakage
If our signal is a lowpass or passband signal, the application of a window will introduce high-frequency components. Power from the original signal will be diverted from the specified frequency band into the high-frequency areas. This redistribution of power from a target band to the upper frequencies is known as leakage.

If we look at a rectangular window, we know from duality that the frequency response of that window is a sinc function, which has non-zero values all the way out to both positive and negative infinity. Convolution of the sinc function with any narrow-band signal is going to cause a very spread-out spectrum.

Hamming
function of hamming

0.54+0.46cos(2*pi*n/N-1)

Blackman window
0.42+.5cos(2*pi*n/N-1)+.08cos(4*pi*n/N-1)

Greatest stop band attenuation of mentioned windowing techniques at the expense of a larger transition band.

The eq. for the symmetric 4-term Blackman Harris window of length N is....

w(n) = a0 - a1*cos(2*pi*n/N-1) + a2*cos(4*pi*n/N-1) - a3*cos(2*pi*n/N-1); 0 <= n <= N-1

a0 = 0.35875   a1 = 0.48829    a2 = 0.14128   a3 = 0.01168